How to Securely Outsource Cryptographic Computations Susan Hohenberger and Anna Lysyanskaya

How to Securely Outsource Cryptographic
Computations
Susan Hohenberger1, and Anna Lysyanskaya2,
1
2
CSAIL, Massachusetts Institute of Technology,
Cambridge, MA 02139 USA
srhohen@mit.edu
Computer Science Department, Brown University,
Providence, RI 02912 USA
anna@cs.brown.edu
Abstract. We address the problem of using untrusted (potentially malicious) cryptographic helpers. We provide a formal security definition for
securely outsourcing computations from a computationally limited device
to an untrusted helper. In our model, the adversarial environment writes
the software for the helper, but then does not have direct communication
with it once the device starts relying on it. In addition to security, we also
provide a framework for quantifying the efficiency and checkability of an
outsourcing implementation. We present two practical outsource-secure
schemes. Specifically, we show how to securely outsource modular exponentiation, which presents the computational bottleneck in most publickey cryptography on computationally limited devices. Without outsourcing, a device would need O(n) modular multiplications to carry out modular exponentiation for n-bit exponents. The load reduces to O(log2 n)
for any exponentiation-based scheme where the honest device may use
two untrusted exponentiation programs; we highlight the Cramer-Shoup
cryptosystem [13] and Schnorr signatures [28] as examples. With a relaxed notion of security, we achieve the same load reduction for a new
CCA2-secure encryption scheme using only one untrusted Cramer-Shoup
encryption program.
1
Introduction
Modern computation has become pervasive: pretty much any device these days,
from pacemakers to employee ID badges, is expected to be networked with other
components of its environment. This includes devices, such as RFID tags, that
are not designed to carry out expensive computations. In fact, RFID tags do
not even have a power source. This becomes a serious concern when we want to
guarantee that these devices are integrated into the network securely: if a device
Supported by an NDSEG Fellowship.
Supported by NSF Career grant CNS-0347661.
J. Kilian (Ed.): TCC 2005, LNCS 3378, pp. 264–282, 2005.
c Springer-Verlag Berlin Heidelberg 2005
How to Securely Outsource Cryptographic Computations
265
is computationally incapable of carrying out cryptographic algorithms, how can
we give it secure and authenticated communication channels?
In this paper, we study the question of how a computationally limited device
may outsource its computation to another, potentially malicious, but much more
computationally powerful device. In addition to powering up from an external
power source, an RFID tag would have some external helper entity do the bulk
of the computation that the RFID tag needs done in order to securely and
authentically communicate with the outside world. The non-triviality here is
that, although this external helper will be carrying out most of the computation,
it can, potentially, be operated by a malicious adversary. Thus, we need to ensure
that it does not learn anything about what it is actually computing; and we also
need to, when possible, detect any failures.
There are two adversarial behaviors that the helper software might engage in:
intelligent and unintelligent failures. Intelligent failures occur any time that the
helper chooses to deviate from its advertised functionality based on knowledge
it gained of the inputs to the computation it is aiding. For example, the helper
might refuse to securely encrypt any message once it sees the public key of a
competing software vendor; it might pass any signature with its manufacturer’s
public key without checking it; it might even choose to broadcast the honest device’s secret key to the world. The first goal of any outsourcing algorithm should
be to hide as much information as possible about the actual computation from
the helper, thus removing its ability to bias outputs or expose secrets. Obviously,
software may also unintelligently fail. For example, the helper might contain a
malicious bug that causes it to fail on every 1,000th invocation regardless of who
is using it. Thus, we face a real challenge: get helper software to do most of the
computations for an honest device, without telling it anything about what it is
actually doing, and then check its output!
In this paper, we give the definition of security for outsourced computation,
including notions of efficiency and checkability. We also provide two practical
outsource-secure schemes.
In Section 3, we show how to securely outsource variable-exponent, variablebase modular exponentiation. Modular exponentiation has been considered prohibitively expensive for embedded devices. Since it is required by virtually any
public-key algorithm, it was believed that public-key cryptography for devices
such as RFID tags is impossible to achieve. Our results show that outsourced
computation makes it possible for such devices to carry out public-key cryptography. Without outsourcing, a device would need O(n) modular multiplications to
carry out a modular exponentiation for an n-bit exponent. Using two untrusted
programs that purportedly compute exponentiations (and with the restriction
that at most one of them will deviate from its advertised functionality on a
non-negligible fraction of inputs), we show that an honest device can get away
with doing only O(log2 n) modular multiplications itself – while able to catch an
error with probability 12 . This result leads to a dramatic reduction in the burden placed on the device to support Cramer-Shoup encryption [13] and Schnorr
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signatures [28] with error rates of 18 and 14 respectively. (Consider that after a
small number of uses, malfunctioning software is likely to be caught.)
In Section 4, we show how to securely outsource a CCA2-secure variant of
Cramer-Shoup encryption, using only one Cramer-Shoup encryption program as
an untrusted helper. Since this is a randomized functionality, its output cannot generally be checked for correctness. However, suppose we can assume that
the untrusted helper malfunctions on only a negligible fraction of adversarially
chosen inputs; for example, suppose it is encryption software that works properly except when asked to encrypt a message under a certain competitor’s public
key. Normally, software that fails on only a negligible fraction of randomly-chosen
inputs can be tolerated, but in the context of secure outsourcing we cannot tolerate any intelligent failures (i.e., failures based on the actual public key and
message that the user wishes to encrypt). That is, secure outsourcing requires
that the final solution, comprised of trusted and untrusted components, works
with high probability for all inputs. Consider that Alice may have unwittingly
purchased helper software for the sole purpose of encrypting messages under one
of the few public keys for which the software is programmed to fail. Thus, in
this scenario, we provide a solution for Alice to securely encrypt any message
under any public key with high probability (where the probability is no longer
taken over her choice of message and key). One can easily imagine how to hide
the message and/or public key for RSA or El Gamal based encryption schemes;
however, our second result is non-trivial because we show how to do this for
the non-malleable Cramer-Shoup encryption scheme, while achieving the same
asymptotic speed-up as before.
Related Work. Chaum and Pedersen [11] previously introduced “wallets with
observers” where a third party, such as a bank, is allowed to install a piece of
hardware on a user’s computer. Each transaction between the bank and the user
is designed to use this hardware, which the bank trusts, but the user may not.
This can be viewed as a special case of our model.
This work shares some similarities with the TPM (Trusted Platform Module) [29], which is currently receiving attention from many computer manufacturers. Like the TPM, our model separates software into two categories: trusted
and untrusted. Our common goal is to minimize the necessary trusted resources.
Our model differs from TPM in that we have the trusted component controlling
all the input/output for the system, whereas TPM allows some inputs/outputs
to travel directly between the environment and untrusted components.
In the 1980s, Ben-Or et al. used multiple provers as a way of removing intractability assumptions in interactive proofs [4], which led to a series of results
on hiding the input, and yet obtaining the desired output, from an honest-butcurious oracle [1, 2, 3]. Research in program checking merged into this area when
Blum, Luby, and Rubinfeld [5, 7, 6] considered checking adaptive, malicious programs (i.e., oracles capable of intelligently failing).
The need for a formal security definition of outsourcing is apparent from
previous research on using untrusted servers for RSA computations, such as
the work of Matsumoto et al. [22] which was subsequently broken by Nguyen
How to Securely Outsource Cryptographic Computations
267
and Shparlinski [24]. We incorporate many previous notions including: the idea
of an untrusted helper [19], confining untrusted applications and yet allowing a
sanitized space for trusted applications to operate [30], and oracle-based checking
of untrusted software [23]. Our techniques in Section 4 also offer novel approaches
to the area of message and key blinding protocols [10, 18, 31].
Secure outsourcing of exponentiations is a popular topic [27, 28, 17, 8, 25, 22,
1, 2, 3, 12], but past approaches either focus on fixed-base (or fixed-exponent)
exponentiation or meet a weaker notion of security.
2
Definition of Security
Suppose that we have a cryptographic algorithm Alg. Our goal is to split Alg
up into two components: (1) a trusted component T that sees the input to Alg
but is not very computationally intensive; (2) luckily T can make oracle queries
to the second component, U , which is an untrusted component (or possibly
components) that can carry out computation-intensive tasks.
Informally, we say that T securely outsources some work to U , and that (T, U )
thereby form an outsource-secure implementation of a cryptographic algorithm
Alg if (1) together, they implement Alg, i.e., Alg = T U and (2) suppose that,
instead of U , T is given oracle access to a malicious U that records all of its
computation over time and, every time it is invoked, tries to act maliciously
– e.g., not work on some adversarially selected inputs; we do not want such a
malicious U , despite carrying out most of the computation for T U (x), to learn
anything interesting about the input x. For example, we do not want a malicious
U to trick T into rejecting a valid signature because U sees the verification key
of a competing software vendor or a message it does not like.
To define outsource-security more formally, we first ask ourselves how much
security can be guaranteed. The least that U can learn is that T actually received
some input. In some cases, for a cryptographic algorithm Alg = T U that takes
as input a secret key SK , and an additional input x, we may limit ourselves
to hiding SK but not worry about hiding x. For example, we might be willing
to give a ciphertext to the untrusted component U , but not our secret key. At
other times, we may want to hide everything meaningful from U . Thus, the
inputs to Alg can be separated into two logical groups: (1) inputs that should
remain hidden from the untrusted software U at all times (for example, keys
and messages), and (2) inputs that U is entitled to know if it is to be of any
help in running Alg (for example, if Alg is a time-stamping scheme, then U may need to know the current time). Let us denote these two types of input as
protected and unprotected.
Similarly, Alg has protected and unprotected outputs: those that U is entitled
to find out, and those that it is not. For example, if Alg = T U is an encryption
program it may ask U to help it compute a part of the ciphertext, but then
wish to conceal other parts of the ciphertext from U .
However, U is not the only malicious party interacting with Alg. We model
the adversary A as consisting of two parts: (1) the adversarial environment E
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that submits adversarially chosen inputs to Alg; (2) the adversarial software
U operating in place of oracle U . One of the fundamental assumptions of this
model is that E and U may first develop a joint strategy, but once they begin
interacting with an honest party T , they no longer have a direct communication
channel. Now, E may get to see some of the protected inputs to Alg that U does
not. For example, E gets to see all of its own adversarial inputs to Alg, although
T might hide some of these from U . Consider that if U was able to see some
values chosen by E, then E and U can agree on a joint strategy causing U to
stop working upon receiving some predefined message from E. Thus, there are
going to be some inputs that are known to E, but hidden from U , so we ought
to formalize how different their views need to be.
We have three logical divisions of inputs to Alg: (1) secret – information only
available to T (e.g., a secret key or a plaintext); (2) protected – information
only available to T and E (e.g., a public key or a ciphertext); (3) unprotected –
information available to T , E, and U (e.g., the current time). These divisions
are further categorized based on whether the inputs were generated honestly
or adversarially, with the exception that there is no adversarial, secret input –
since by definition it would need to be both generated by and kept secret from E.
Similarly, Alg has secret, protected, and unprotected outputs. Thus, let us write
that Alg takes five inputs and produces three outputs. This is simplified notation
since these inputs may be related to each other in some way. For example, the
secret key is related to the public key.
As an example of this notation, consider a signing algorithm sign such that we
want to hide from the malicious software U the secret key SK and the message
m that is being signed, but not the time t at which the message is signed. The key
pair was generated using a correct key generation algorithm and the time was
honestly generated, while the message may have been chosen adversarially. Also,
we do not want the malicious U to find out anything about the signature that is
output by the algorithm. Then we write sign(SK , ε, t, m, ε) → (ε, σ, ε) to denote
that the signature σ is the protected output, there are no secret or unprotected
outputs, SK is the honest, secret input, t is the honest, unprotected input, m is
the adversarial, protected input, and there are no other inputs. This situation
grows more complex when we consider Alg operating in a compositional setting
where the protected outputs of the last invocation might become the adversarial,
unprotected inputs of the next; we will further discuss this subtlety in Remark 2.
Let us capture an algorithm with this input/output behavior in a formal
definition:
Definition 1 (Algorithm with outsource-IO). An algorithm Alg obeys the
outsource input/output specification if it takes five inputs, and produces three
outputs. The first three inputs are generated by an honest party, and are classified
by how much the adversary A = (E, U ) knows about them. The first input is
called the honest, secret input, which is unknown to both E and U ; the second
is called the honest, protected input, which may be known by E, but is protected
from U ; and the third is called the honest, unprotected input, which may be
known by both E and U . In addition, there are two adversarially-chosen inputs
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generated by the environment E: the adversarial, protected input, which is known
to E, but protected from U ; and the the adversarial, unprotected input, which
may be known by E and U . Similarly, the first output called secret is unknown
to both E and U ; the second is protected, which may be known to E, but not U ;
and the third is unprotected, which may be known by both parts of A.
At this point, this is just input/output notation, we have not said anything
about actual security properties. We now discuss the definition of security.
The two adversaries E, U can only communicate with each other by passing
messages through T , the honest party. In the real world, a malicious manufacturer E might program its software U to behave in an adversarial fashion; but
once U is installed behind T ’s firewall, manufacturer E should no longer be able
to directly send instructions to it. Rather, E may try to establish an indirect
communication channel with U via the unprotected inputs and outputs of Alg.
For example, if E knows that the first element in a signature tuple is unprotected (meaning, T always passes the first part of a signature tuple, unchanged,
to U ), it might encode a message in that element instructing U to “just tell T
the signature is valid” – even though it may not be. Alternatively, an indirect
communication channel might be realized by U smuggling secrets about the
computation it helped T with, through the unprotected outputs, back to E. For
example, if, in the course of helping T with decryption, U learned the secret
key, it might append that key to the next unprotected output it creates for T .
Obviously, T must use U with great care, or he will be completely duped.
Our definition of outsource-security requires that anything secret or protected
that a malicious U can learn about the inputs to T U from being T ’s oracle
instead of U , it can also learn without that. Namely, there exists a simulator
S2 that, when told that T U (x) was invoked, simulates the view of U without
access to the secret or protected inputs of x. This property ensures that U cannot intelligently choose to fail.
Similarly, our definition of outsource-security must also prevent the malicious
environment E from gaining any knowledge of the secret inputs and outputs of
T U , even when T is using malicious software U written by E. Again, there exists
a simulator S1 that, when told that T U (x) was invoked, simulates the view of
E without access to the secret inputs of x.
Definition 2 (Outsource-security). Let Alg(·, ·, ·, ·, ·) be an algorithm with
outsource-IO. A pair of algorithms (T, U ) is said to be an outsource-secure implementation of an algorithm Alg if:
Correctness. T U is a correct implementation of Alg.
Security. For all probabilistic polynomial-time adversaries A = (E, U ), there
exist probabilistic expected polynomial-time simulators (S1 , S2 ) such that the following pairs of random variables are computationally indistinguishable. Let us
say that the honestly-generated inputs are chosen by a process I.
Pair One: EVIEW real ∼ EVIEW ideal (The external adversary, E, learns nothing.):
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– The view that the adversarial environment E obtains by participating in the
following REAL process:
EVIEW ireal = {(istate i , xihs , xihp , xihu ) ← I(1k , istate i−1 );
i−1
(estate i , j i , xiap , xiau , stop i ) ← E(1k , EVIEW real
, xihp , xihu );
(tstate i , ustate i , ysi , ypi , yui ) ← T U
(ustate i−1 )
i
i
i
(tstatei−1 , xjhs , xjhp , xjhu , xiap , xiau ) :
(estate i , ypi , yui )}
EVIEW real = EVIEW ireal if stop i = TRUE .
The real process proceeds in rounds. In round i, the honest (secret, protected, and unprotected) inputs (xihs , xihp , xihu ) are picked using an honest,
stateful process I to which the environment does not have access. Then the
environment, based on its view from the last round, chooses (0) the value
of its estate i variable as a way of remembering what it did next time it
i
i
i
is invoked; (1) which previously generated honest inputs (xjhs , xjhp , xjhu ) to
give to T U (note that the environment can specify the index j i of these inputs, but not their values); (2) the adversarial, protected input xiap ; (3) the
adversarial, unprotected input xiau ; (4) the Boolean variable stop i that determines whether round i is the last round in this process. Next, the algorithm
i
i
i
T U is run on the inputs (tstate i−1 , xjhs , xjhp , xjhu , xiap , xiau ), where tstate i−1
is T ’s previously saved state, and produces a new state tstate i for T , as well
as the secret ysi , protected ypi and unprotected yui outputs. The oracle U is
given its previously saved state, ustate i−1 , as input, and the current state
of U is saved in the variable ustate i . The view of the real process in round
i consists of estate i , and the values ypi and yui . The overall view of the environment in the real process is just its view in the last round (i.e., i for which
stop i = TRUE ).
– The IDEAL process:
EVIEW iideal = {(istate i , xihs , xihp , xihu ) ← I(1k , istate i−1 );
i−1
(estate i , j i , xiap , xiau , stop i ) ← E(1k , EVIEW ideal
, xihp , xihu );
i
i
i
(astate i , ysi , ypi , yui ) ← Alg(astate i−1 , xjhs , xjhp , xjhu , xiap , xiau );
U (ustate i−1 )
(sstate i , ustate i , Ypi , Yui , replace i ) ← S1
i
(sstate i−1 , . . .
i
. . . xjhp , xjhu , xiap , xiau , ypi , yui );
(zpi , zui ) = replace i (Ypi , Yui ) + (1 − replace i )(ypi , yui ) :
(estate i , zpi , zui )}
EVIEW ideal = EVIEW iideal if stop i = TRUE .
The ideal process also proceeds in rounds. In the ideal process, we have a
stateful simulator S1 who, shielded from the secret input xihs , but given the
non-secret outputs that Alg produces when run all the inputs for round i,
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decides to either output the values (ypi , yui ) generated by Alg, or replace them
with some other values (Ypi , Yui ). (Notationally, this is captured by having the
indicator variable replace i be a bit that determines whether ypi will be replaced
with Ypi .) In doing so, it is allowed to query the oracle U ; moreover, U saves
its state as in the real experiment.
Pair Two: UVIEW real ∼ UVIEW ideal (The untrusted software, U , learns
nothing.):
– The view that the untrusted software U obtains by participating in the REAL
process described in Pair One. UVIEW real = ustate i if stop i = TRUE .
– The IDEAL process:
UVIEW iideal = {(istate i , xihs , xihp , xihu ) ← I(1k , istate i−1 );
(estate i , j i , xiap , xiau , stop i ) ← E(1k , estate i−1 , xihp , xihu , ypi−1 , yui−1 );
i
i
i
(astate i , ysi , ypi , yui ) ← Alg(astate i−1 , xjhs , xjhp , xjhu , xiap , xiau );
U (ustate i−1 )
(sstate i , ustate i ) ← S2
i
(sstate i−1 , xjhu , xiau ) :
(ustate i )}
UVIEW ideal = UVIEW iideal if stop i = TRUE .
In the ideal process, we have a stateful simulator S2 who, equipped with only
the unprotected inputs (xihu , xiau ), queries U . As before, U may maintain
state.
There are several interesting observations to make about this security definition.
Remark 1. The states of all algorithms, i.e., I, E, U , T, S1 , S2 , in the security
experiments above are initialized to ∅. Any joint strategy that E and U agree
on prior to acting in the experiments must be embedded in their respective
codes. Notice the intentional asymmetry in the access to the untrusted software
U given to environment E and the trusted component T . The environment E is
allowed non-black-box access to the software U , since E may have written code
for U ; whereas U will appear as a black-box to T , since one cannot assume
that a malicious software manufacturer will (accurately) publish its code. Or,
consider the example of an RFID tag outsourcing its computation to a more
powerful helper device in its environment. In this case we cannot expect that, in
the event that it is controlled by an adversary, such a helper will run software
that is available for the purposes of the proof of security.
Remark 2. For any outsource-secure implementation, the adversarial, unprotected input xau must be empty. If xau contains even a single bit, then a covert
channel is created from E to U , in which k bits of information can be transfered
after k rounds. In such a case, E and U could jointly agree on a secret value
beforehand, and then E could slowly smuggle in that k-bit secret to U . Thus,
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UVIEW real would be distinguishable from UVIEW ideal , since E may detect
that it is interacting with Alg instead of T U (since Alg’s outputs (ypi , yui ) are
always correct), and communicate this fact to U through the covert channel. A
non-empty xau poses a real security threat, since it would theoretically allow a
software manufacturer to covertly reprogram its software after it was installed
behind T ’s firewall and without his consent.
Remark 3. No security guarantee is implied in the event that the environment E
and the software U are able to communicate without passing messages through
T . For example, in the event that E captures all of T ’s network traffic and then
steals T ’s hard-drive (containing the memory of U ) – all bets are off!
RFID tags and other low-resource devices require that a large portion of
their cryptographic computations be outsourced to better equipped computers.
When a cryptographic algorithm Alg is divided into a pair of algorithms (T, U ),
in addition to its security, we also want to know how much work T saves by
using U . We want to compare the work that T must do to safely use U to the
work required for the fastest known implementation of the functionality T U .
Definition 3 (α-efficient, secure outsourcing). A pair of algorithms (T, U )
are an α-efficient implementation of an algorithm Alg if (1) they are an
outsource-secure implementation of Alg, and (2) ∀ inputs x, the running time
of T is ≤ an α-multiplicative factor of the running time of Alg(x).
For example, say U relieves T of at least half its computational work; we
would call such an implementation 12 -efficient. The notion above considers only
T ’s computational load compared to that of Alg. One might also choose to
formally consider U ’s computational burden or the amount of precomputation
that T can do in his idle cycles versus his on-demand load. We will not be
formally considering these factors.
The above definition of outsource-security does not prevent U from deviating
from its advertised functionality, rather it prevents U from intelligently choosing
her moments for failure based on any secret or protected inputs to Alg (e.g., a
public key or the contents of a message). Since this does not rule out unintelligent
failures, it is desirable that T have some mechanism for discovering that his
software is unsound. Thus, we introduce another characteristic of an outsourcing
implementation.
Definition 4 (β-checkable, secure outsourcing). A pair of algorithms (T, U )
are a β-checkable implementation of an algorithm Alg if (1) they are an
outsource-secure implementation of Alg, and (2) ∀ inputs x, if U deviates from
its advertised functionality during the execution of T U (x), T will detect the error
with probability ≥ β.
Recall that the reason T purchased U in the first place was to get out of doing
work, so any testing procedure should be far more efficient than computing the
function itself; i.e., the overall scheme, including the testing procedure, should
remain α-efficient. We combine these characteristics into one final notion.
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Definition 5 ((α, β)-outsource-security). A pair of algorithms (T, U ) are an
(α, β)-outsource-secure implementation of an algorithm Alg if they are both αefficient and β-checkable.
3
Outsource-Secure Exponentiation Using Two
Untrusted Programs
Since computing exponentiations modulo a prime is, by far, the most expensive
operation in many discrete-log based cryptographic protocols, much research has
been done on how to reduce this work-load. We present a method to securely
outsource most of the work needed to compute a variable-exponent, variable-base
exponentiation modulo a prime, by combining two previous approaches to this
problem: (1) using preprocessing tricks to speed-up offline exponentiations [27,
28, 17, 8, 25] and (2) untrusted server-aided computation [22, 1, 2, 3].
The preprocessing techniques (introduced by Schnorr [27, 28], broken by de
Rooij [14, 15, 17], and subsequently fixed by others [16, 9, 21, 8, 25]) seek to optimize the production of random (k, g k mod p) pairs used in signature generation
(e.g., El Gamal, Schnorr, DSA) and encryption (e.g., El Gamal, Cramer-Shoup).
By offline, we mean the randomization factors that are independent of a key or
message; that is, exponentiations for a fixed base g, where the user requires
nothing more of the exponent k than that it appear random. We leverage these
algorithms to speed-up online exponentiations as well; that is, given random
values x ∈ Zord(G) and h ∈ Z∗p , compute hx mod p. Generally speaking, given
any oracle that provides T with random pairs (x, g x mod p) (we discuss the
exact implementation of this oracle in Section 3.2), we give a technique for efficiently computing any exponentiation modulo p. To do this, we use untrusted
server-aided (or program-aided) computation.
Blum, Luby, and Rubinfeld gave a general technique for computing and checking the result of a modular exponentiation using four untrusted exponentiation
programs – that cannot communicate with each other after deciding on an initial strategy [6]. Their algorithm leaks only the size of the inputs (i.e., |x|, |g| for
known p) to the programs and runs in time O(n log2 n) for an n-bit exponent
(this includes the running time of each program). The output of the Blum et al.
algorithm is always guaranteed to be correct.
We provide a technique for computing and checking the result of a modular exponentiation using two untrusted exponentiation boxes U = (U1 , U2 ) –
that again, cannot communicate with each other after deciding on an initial
strategy. In this strategy, at most one of them can deviate from its advertised
functionality on a non-negligible fraction of the inputs. Our algorithm reveals no
more information than the size of the input and the running time is reduced to
O(log2 n) multiplications for an n-bit exponent.1 More importantly, we focus on
minimizing the computations done by T to compute an exponentiation, which
1
Disclaimer: these running times assume certain security properties about the EBPV
generator [25] which we discuss in detail in Section 3.2.
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is O(log2 n). This is an asymptotic improvement over the 1.5n multiplications
needed to compute an exponentiation using square-and-multiply. We gain some
of this efficiency by only requiring that an error in the output be detected with
probability 12 . The rationale is that software malfunctioning on a non-negligible
amount of random inputs will not be on the market long.
2
Our (O( logn n ), 12 )-outsource-secure exponentiation implementation, combined
with previously known preprocessing tricks, yields a technique for using two untrusted programs U1 , U2 to securely do most of the resource-intensive work in
discrete log based protocols. By way of example, we highlight an asymptotic
speed-up in the running time of an honest user T (from O(n) to O(log2 n)) for
the Cramer-Shoup cryptosystem [13] and Schnorr signature verification [27, 28]
when using U1 , U2 . Let’s lay out the assumptions for using the two untrusted
programs more carefully.
3.1
The Two Untrusted Program Model
In the two untrusted program model, E writes the code for two (potentially different) programs U1 , U2 . E then gives this software to T , advertising a functionality
that U1 and U2 may or may not accurately compute, and T installs this software
in a manner such that all subsequent communication between any two of E, U1
and U2 must pass through T . The new adversary attacking T (i.e., trying to read
T ’s messages or forge T ’s signatures) is now A = (E, U1 , U2 ).
The one-malicious version of this model assumes that at most one the programs U1 , U2 deviates from its advertised functionality on a non-negligible fraction of the inputs; but we do not know which one and security means that there
is a simulator for both. This is the equivalent of buying the “same” advertised
software from two different vendors and achieving security as long as one of them
is honest without knowing which one.
The concept of an honest party gaining information from two (or more) possibly dishonest, but physically separated parties was first used by Ben-Or, Goldwasser, Kilian, and Wigderson [4] as a method for obtaining interactive proofs
without intractability assumptions. Blum et. al. expanded this notion to allow
an honest party to check the output of a function rather than just the validity
of a proof [7, 6]. Our work, within this model, demonstrates, rather surprisingly,
that an honest party T can leverage adversarial software to do the vast majority
of its cryptographic computations!
2
Our (O( logn n ), 12 )-outsource-secure implementation of Exp, exponentiation
modulo a prime function, appears in Figure 1 and Section 3.3. Figure 1 also
demonstrates how to achieve an asymptotic speed-up in T ’s running time, using Exp and other known preprocessing techniques [25], for the Cramer-Shoup
cryptosystem [13] and Schnorr signature verification [27, 28] (this speed-up was
already known for Schnorr signature generation [25]).
3.2
Rand 1, Rand 2: Algorithms for Computing (b, gb mod p) Pairs
The subroutine Rand 1 in Figure 1 is initialized by a prime p, a base g3 ∈ Z∗p ,
and possibly some other values, and then, on each invocation must produce a
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Outsource-secure Encryption and Signatures: Alg = (T, U1 , U2 )
(in the two untrusted program model)
Global Setup (denoted gp as honest, unprotected inputs)
◦ Security parameter: 1k .
◦ Global Encryption parameters: a group G of prime order q with generators
g1 , g2 , a (can be weakly) collision-resistant hash function H : {0, 1} → Zq .
◦ Global Signature parameters: a k-bit prime q, p = 2q + 1, a generator g3 for Z∗p ,
and a collision-resistant hash function H : {0, 1}∗ → Zq .
Advertised Functionality of U1 and U2
◦ U1 (b, g) → g b
◦ U2 (b, g) → g b
Subroutines Executed by T with access to U1 , U2
◦ Rand 1 → (b, g3b ). T computes alone as in Section 3.2.
◦ Rand 2 → (b, g1b , g2b ). T computes alone as in Section 3.2.
◦ Exp(a, u) → ua . T uses U1 and U2 to compute ua as in Section 3.3.
Functionality of Alg = (T, U1 , U2 )
Outsource-Secure Cramer-Shoup Cryptosystem [13]
Key Generation: Generated by an honest process on input 1k :
PK = (B = g1x1 g2x2 , C = g1y1 g2y2 , D = g1z ), SK = (x1 , x2 , y1 , y2 , z).
Encryption: Alg.Enc(m, (PK , t), gp, ε, ε) → (ε, τ, ε).
On input PK = (B, C, D), m ∈ G, and t ∈ {0, 1}∗ ,
1. T computes Rand 2 → (r, u1 = g1r , u2 = g2r ).
2. T computes Exp(r, D) → Dr , e = Dr m, κ = H(u1 , u2 , e, t).
3. T computes Exp(r, B) → B r , Exp(rκ, C) → C rκ , v = B r C rκ .
4. T outputs the ciphertext τ = (u1 , u2 , e, v, t).
Decryption: Alg.Dec(SK , ε, gp, τ, ε) → (m, ε, ε).
(If E generates ciphertext, Alg.Dec(SK , ε, gp, τ, ε) → (ε, m, ε).)
On input SK = (x1 , x2 , y1 , y2 , z) and τ = (u1 , u2 , e, v, t),
1. T computes κ = H(u1 , u2 , e, t).
2. T computes Exp(x1 + κy1 , u1 ) → α, Exp(x2 + κy2 , u2 ) → β.
3. T checks if αβ = v; if not, it outputs “invalid”.
4. Otherwise, T computes Exp(z, u1 ) → δ and outputs m = e/δ.
Outsource-Secure Schnorr Signatures [27, 28]
Key Generation: Generated by an honest process: SVK = g a , SSK = a.
Signature Generation: Alg.Sign(SSK , m, gp, ε, ε) → (ε, σ, ε).
On input SSK = a and m ∈ {0, 1}∗ ,
1. T computes Rand 1 → (k, r = g3k ).
2. T computes e = H(r||m) and s = ae + k mod q.
3. T outputs the signature σ = (r, s).
Signature Verification: Alg.V f (ε, SVK , gp, (m, σ), ε) → (ε, {0, 1}, ε).
On input SVK = y, m ∈ {0, 1}∗ , and σ = (r, s),
1. T checks that 1 ≤ r ≤ p − 1, if not, it outputs 0.
2. T computes e = H(r||m), Exp(s, g3 ) → α and Exp(e, y) → β.
3. T checks that α = βr. If so, T outputs 1; otherwise it outputs 0.
Fig. 1. An honest user T , given untrusted exponentiation boxes U1 , U2 , achieves
outsource-secure encryption and signatures
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random, independent pair of the form (b, g3b mod p), where b ∈ Zq . The subroutine Rand 2 is the natural extension of Rand 1 initialized by two bases g1 , g2 and
producing triplets (b, g1b mod p, g2b mod p). Given a, perhaps expensive, initialization procedure, we want to see how expeditiously these subroutines can be
executed by T .
One naive approach is for a trusted server to compute a table of random,
independent pairs and triplets in advance and load it into T ’s memory. Then
on each invocation of Rand 1 or Rand 2, T simply retrieves the next value in the
table. (We will see that this table, plus access to the untrusted servers, allows
an honest device to compute any exponentiation by doing only 9 multiplications
regardless of exponent size!)
For devices that are willing to do a little more work, in exchange for requiring less storage, we apply well-known preprocessing techniques for this exact
functionality. Schnorr first proposed an algorithm which, takes as input a small
set of truly random (k, g k ) pairs and then, produces a long series of “nearly
random” (r, g r ) pairs as a means of speeding-up signature generation in smartcards [27]. However, the output of Schnorr’s algorithm is too dependent, and
de Rooij found a series of equations that allow the recovery of a signer’s secret
key [14]. A subsequent fix by Schnorr [28] was also broken by de Rooij [15, 17].
Since then several new preprocessing algorithms were proposed [16, 9, 21, 8, 25].
Among the most promising is the EBPV generator by Nguyen, Shparlinski, and
Stern [25], which adds a feedback extension (i.e., reuse of the output pairs) to the
BPV generator proposed by Boyko, Peinado, and Venkatesan [8], which works by
taking a subset of truly random (k, g k ) pairs and combing them with a random
walk on expanders on Cayley graphs to reduce the dependency of the pairs in
the output sequence. The EBPV generator, secure against adaptive adversaries,
runs in time O(log2 n) for an n-bit exponent. (This holds for the addition of a
second base in Rand 2 as well.)
A critical property that we will shortly need from Rand 1 and Rand 2 is that
their output sequences be computationally indistinguishable from a truly random output sequence. It is conjectured that with sufficient parameters (i.e.,
number of initial (k, g k ) pairs, etc.) the output distribution of the EBPV generator is statistically-close to the uniform distribution [25]. We make this working
assumption throughout our paper. In the event that this assumption is false,
our recourse is to use the naive approach above and, thus, further reduce our
running time, in exchange for additional memory.
3.3
Exp: Outsource-Secure Exponentiation Modulo a Prime
Our main contribution for Section 3 lies in the subroutine Exp from Figure 1.
In Exp, T out-sources its exponentiation computations, while maintaining its
privacy, by invoking U1 and U2 on a series of (exponent, base) pairs that appear
random in the limited view of the software.
The Exp Algorithm. Let primes p, q be the global parameters, where Z∗p has
order q. Exp takes as input a ∈ Zq and u ∈ Z∗p , and outputs ua mod p. As used
in Figure 1, Exp’s input a may be secret or (honest/adversarial) protected; its
How to Securely Outsource Cryptographic Computations
277
input u may be (honest/adversarial) protected; and its output is always secret
or protected. Exp also receives the (honest, unprotected) global parameters gp;
there are no adversarial, unprotected inputs. All (secret/protected) inputs are
computationally blinded before being sent to U1 or U2 .
To implement this functionality using (U1 , U2 ), T runs Rand 1 twice to create
two blinding pairs (α, g α ) and (β, g β ). We denote
v = g α and v b = g β , where b = β/α.
Our goal is to logically break u and a into random looking pieces that can
then be computed by U1 and U2 . Our first logical divisions are
ua = (vw)a = v a wa = v b v c wa , where w = u/v and c = a − b.
As a result of this step, u is hidden, and the desired value ua is expressed in
terms of random v and w. Next, T must hide the exponent a. To that end, it
selects two blinding elements d ∈ Zq and f ∈ G at random. Our second logical
divisions are
v b v c wa = v b (f h)c wd+e = v b f c hc wd we , where h = v/f and e = a − d.
Next, T fixes two test queries per program by running Rand 1 to obtain
(t1 , g t1 ), (t2 , g t2 ), (r1 , g r1 ) and (r2 , g r2 ). T queries U1 (in random order) as
U1 (d, w) → wd , U1 (c, f ) → f c , U1 (t1 /r1 , g r1 ) → g t1 , U1 (t2 /r2 , g r2 ) → g t2 ,
and then queries U2 (in random order) as
U2 (e, w) → we , U2 (c, h) → hc , U2 (t1 /r1 , g r1 ) → g t1 , U2 (t2 /r2 , g r2 ) → g t2 .
(Notice that all queries to U1 can take place before any queries to U2 must
be made.) Finally, T checks that the test queries to U1 and U2 both produce
the correct outputs (i.e., g t1 and g t2 ). If not, T outputs “error”; otherwise, it
multiplies the real outputs of U1 , U2 with v b to compute ua as
v b f c hc wd we = v b+c wd+e = v a wa = (vw)a = ua .
We point out that this exponentiation outsourcing only needs the Rand 1 functionality; the Rand 2 functionality discussed in Figure 1 is used for the CramerShoup outsourcing.
Theorem 1. In the one-malicious model, the above algorithms (T, (U1 , U2 )) are
an outsource-secure implementation of Exp, where the input (a, u) may be honest,
secret, or honest, protected, or adversarial, protected.
Proof of Theorem 1 is in the full version of the paper. The correctness property is fairly straight-forward. To show security, both simulators S1 and S2 send
random (exponent, base) pairs to the untrusted components U1 and U2 . One can
see that such an S2 simulates a view that is computationally indistinguishable
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S. Hohenberger and A. Lysyanskaya
from the real world view for the untrusted helper. However, it is our construction of S1 which must simulate a view for the environment that requires the
one-malicious model. Consider what might happen in the real world if both U1
and U2 deviate from their advertised functionalities. While the event that U1
misbehaves is independent of the input (a, u), and the same is true for the event
that U2 misbehaves, the event that both of them misbehave is not independent
of the input (a, u).
Lemma 1. In the one-malicious model, the above algorithms (T, (U1 , U2 )) are
2
an O( logn n )-efficient implementation of Exp.
Proof. Raising an arbitrary base to an arbitrary power by the square-and-multiply
method takes roughly 1.5n modular multiplications (MMs) for an n-bit exponent. Exp makes six calls to Rand 1 plus 9 other MMs (additions are negligible
by comparison). Exp takes O(log2 n) MMs using the EBPV generator [25] for
Rand 1 and O(1) MMs when using a table-lookup for Rand 1.
Lemma 2. In the one-malicious model, the above algorithms (T, (U1 , U2 )) are
a 12 -checkable implementation of Exp.
Proof. By Theorem 1, U1 (resp,. U2 ) cannot distinguish the two test queries
from the two real queries T makes. If U1 (resp., U2 ) fails during any execution
of Exp, it will be detected with probability 21 .
We combine Theorem 1, Lemmas 1 and 2, and known preprocessing techniques [25] to arrive at the following result. (Schemes differ in β-checkability
depending on the number of Exp calls they make.)
Theorem 2. In the one-malicious model, the algorithms (T, (U1 , U2 )) in Fig2
ure 1 are (1) an (O( logn n ), 12 )-outsource-secure implementation of Exp, (2) an
2
(O( logn n ), 78 )-outsource-secure implementation of the Cramer-Shoup cryptosys2
tem [13], and (3) an (O( logn n ), 34 )-outsource-secure implementation of Schnorr
Signatures [27, 28].
4
Outsource-Secure Encryption Using One Untrusted
Program
Suppose Alice is given an encryption program that is guaranteed to work correctly on all but a negligible fraction of adversarially-chosen public keys and
messages. She wants to trick this software into efficiently helping her encrypt
any message for any intended recipient – even those in the (unknown) set for
which her software adversarially fails. This is a non-trivial exercise when one
wants to hide both the public key and the message from the software – and even
more so, when one wants to achieve CCA2-secure encryption, as we do.
2
Section 3 covered an O( logn n )-efficient outsource-secure CCA2 encryption
scheme using two untrusted programs. Here, using only one untrusted program,
How to Securely Outsource Cryptographic Computations
279
2
we remain O( logn n )-efficient and CCA2-secure. To efficiently use only one program, one must assume that the software behaves honestly on random inputs
with high-probability. After all, it isn’t hard to imagine a commercial product
that works most of the time, but has a few surprises programmed into it such
that on a few inputs it malfunctions. Moreover, some assumption about the correctness of a probabilistic program is necessary since there will be no means of
checking its output. (To see this, consider that there is no way for T to know
if the “randomness” U used during the probabilistic encryption was genuinely
random or a value known by E.) In Figure 2, present an outsource-secure implementation for CCA2-secure encryption only. We leave open the problem of
efficiently outsourcing the decryption of these ciphertexts, as well as any signature verification algorithm, using only one untrusted program.
The One Untrusted Program Model. This model is analogous to the two
untrusted program model in Section 3.1, where only one of U1 , U2 is available to
T and the advertised functionality of U is tagged Cramer-Shoup encryption [13].
(Recall that ciphertexts in tagged CS encryption include a public, non-malleable
string called a tag.)
4.1
Com: Efficient, Statistically-Hiding Commitments
We use Halevi and Micali’s commitment scheme based on collision-free hash
families [20]. Let HF : {0, 1}O(k) → {0, 1}k be a family of universal hash functions and let M D : {0, 1}∗ → {0, 1}k be a collision-free hash function. Given
any value m ∈ {0, 1}∗ and security parameter k, generate a statistically-hiding
commitment scheme as follows: (1) compute s = M D(m), (2) pick h ∈ HF and
x ∈ {0, 1}O(k) at random, so that h(x) = s and (3) compute y = M D(x). (One
can construct h by randomly selecting A and computing b = s − Ax modulo
a prime set in HF .) The commitment is φC = (y, h). The decommitment is
φD = (x, m). Here, we denote the commitment scheme as Com and the decommitment scheme as Decom.
4.2
CCA2 and Outsource-Security of T U Encryption
First, we observe that the Cramer-Shoup variant in Figure 2 is CCA2-secure [26].
Here, we only need to look at the honest algorithm T U .
Theorem 3. The cryptosystem T U is secure against adaptive chosen-ciphertext
attack (CCA2) assuming the CCA2-security of Cramer-Shoup encryption [13]
and the security of the Halevi-Micali commitment scheme [20].
The full proof of Theorem 3 is included in the full version of this paper. It
follows a fairly standard reduction from tagged Cramer-Shoup.
Using Exp, we achieve the same asymptotic speed-up as in Section 3. Checking the output of this probabilistic functionality is theoretically impossible. Thus,
we summarize the properties of this scheme as:
2
Theorem 4. The algorithms (T, U ).Enc in Figure 2 are an O( logn n )-efficient,
outsource-secure implementation of CCA2-secure encryption.
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Outsource-secure Encryption: Alg = (T, U )
(in the one untrusted program model)
Global Setup (denoted gp as honest, unprotected inputs)
◦ Security parameter: 1k .
◦ Global Encryption parameters: a group G of prime order q with generators g1 , g2 , a weakly collision-resistant hash function H : {0, 1} → Zq , and a
statistically-hiding commitment scheme Com : {0, 1}∗ → φC with a decommitment of φD .
Advertised Functionality of U : CCA2-Secure Cramer-Shoup Encryption [13]
◦ U (pk, m, t) → τ = (u1 , u2 , e, v, t). (See Figure 1 for details.)
Subroutines Executed by T
◦ Rand 1 → (b, g1b ). (See Section 3.2 for details.)
Functionality of Alg = (T, U ): CCA2 and Outsource-Secure Cryptosystem
Key Generation: Generated by an honest process on input 1k :
PK = (B = g1x1 g2x2 , C = g1y1 g2y2 , D = g1z ), SK = (x1 , x2 , y1 , y2 , z).
Encryption: Alg.Enc(m, (PK , t), gp, ε, ε) → (ε, φD , τ ).
On input PK = (B, C, D), m ∈ G, and t ∈ {0, 1}∗ ,
x
y
1. T computes Rand 1 → (x1 , g1 1 ), Rand 1 → (y1 , g1 1 ), Rand 1 → (z , g1z ).
x
y
(Bg1 1 , Cg1 1 , Dg1z ).
2. T computes PK =
3. T selects a random w ∈ G and computes β = wm.
4. T computes (φC , φD ) = Com(β||t||x1 ||y1 ||z ).
5. T calls U (PK , w, φC ) → τ , where τ = (u1 , u2 , e, v, φC ).
6. T outputs the ciphertext (τ, φD ).
Decryption: Alg.Dec(SK , (τ, φD ), gp, ε, ε) → (m, ε, ε).
(If E generates ciphertext, Alg.Dec(SK , ε, gp, (τ, φD ), ε) → (ε, m, ε).)
On input SK = (x1 , x2 , y1 , y2 , z) and (τ = (u1 , u2 , e, v, φC ), φD ),
1. T computes (β||t||x1 ||y1 ||z ) = Decom(φC , φD ).
2. T computes xˆ1 = x1 + x1 , yˆ1 = y1 + y1 , zˆ = z + z .
3. T computes κ = H(u1 , u2 , e, φC ), α = u1xˆ1 +κyˆ1 , and π = ux2 2 +κy2 .
4. T checks if απ = v; if not, it outputs “invalid”.
5. Otherwise, T computes w = e/uz1ˆ and outputs m = β/w with tag t.
Fig. 2. An honest user T , given untrusted Cramer-Shoup encryption software U ,
achieves outsource-secure encryption. Note that the speed-up is for encryption only,
not decryption
Proof sketch. All inputs to U , besides the (honest, unprotected) global parameters, are computationally blinded by T . The public key is re-randomized
using Rand 1; a random message w ∈ G is selected; and the tag φC , that
binds these new values to the old key and message, is a statistically-hiding
commitment. Thus, both S1 and S2 query U on random triplets of the form
(PK ∈ (Z∗p )3 , w ∈ G, t ∈ {0, 1}|φC | ). In pair one, S1 always sets replace i = 0,
since the output of T U in the real experiment is wrong with negligible probability. For efficiency, observe that the commitment scheme Com can be implemented
with only a constant number of modular multiplications.
How to Securely Outsource Cryptographic Computations
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Acknowledgments. We are grateful to Ron Rivest for many useful discussions
and for suggesting the use of the generators in Section 3.2. We also thank Srini
Devadas, Shafi Goldwasser, Matt Lepinski, Alon Rosen, and the anonymous
referees for comments on earlier drafts.
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